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Question-150517




Question Number 150517 by ajfour last updated on 13/Aug/21
Commented by ajfour last updated on 13/Aug/21
Cannot OP be a bit greater  than OB=R (radius of hemi-  sphere)? If yes then for what  range of semi-vertical ∠ θ  of cone (right), is that so?
$${Cannot}\:{OP}\:{be}\:{a}\:{bit}\:{greater} \\ $$$${than}\:{OB}={R}\:\left({radius}\:{of}\:{hemi}-\right. \\ $$$$\left.{sphere}\right)?\:{If}\:{yes}\:{then}\:{for}\:{what} \\ $$$${range}\:{of}\:{semi}-{vertical}\:\angle\:\theta \\ $$$${of}\:{cone}\:\left({right}\right),\:{is}\:{that}\:{so}? \\ $$
Commented by ajfour last updated on 13/Aug/21
THANKS   SiR. very convincing!
$$\mathcal{THANKS}\:\:\:\mathcal{S}{i}\mathcal{R}.\:{very}\:{convincing}! \\ $$
Commented by mr W last updated on 13/Aug/21
the circle with radius DB (=DP)  is always smaller than the  cross−section circle between the  sphere and the plane in which the  circle with radius DB lies. than means  no point on the circle with radius  DB lies outside the sphere. i.e. P  can never lie outside the sphere, or  OP≤R.
$${the}\:{circle}\:{with}\:{radius}\:{DB}\:\left(={DP}\right) \\ $$$${is}\:{always}\:{smaller}\:{than}\:{the} \\ $$$${cross}−{section}\:{circle}\:{between}\:{the} \\ $$$${sphere}\:{and}\:{the}\:{plane}\:{in}\:{which}\:{the} \\ $$$${circle}\:{with}\:{radius}\:{DB}\:{lies}.\:{than}\:{means} \\ $$$${no}\:{point}\:{on}\:{the}\:{circle}\:{with}\:{radius} \\ $$$${DB}\:{lies}\:{outside}\:{the}\:{sphere}.\:{i}.{e}.\:{P} \\ $$$${can}\:{never}\:{lie}\:{outside}\:{the}\:{sphere},\:{or} \\ $$$${OP}\leqslant{R}. \\ $$
Commented by mr W last updated on 13/Aug/21
Commented by mr W last updated on 13/Aug/21
blue shaded circle and red circle are  in the same plane. red circle is on the  sphere. since blue circle is inside the  red circle. points on blue circle can  never exceed the sphere. ⇒OP≤R.
$${blue}\:{shaded}\:{circle}\:{and}\:{red}\:{circle}\:{are} \\ $$$${in}\:{the}\:{same}\:{plane}.\:{red}\:{circle}\:{is}\:{on}\:{the} \\ $$$${sphere}.\:{since}\:{blue}\:{circle}\:{is}\:{inside}\:{the} \\ $$$${red}\:{circle}.\:{points}\:{on}\:{blue}\:{circle}\:{can} \\ $$$${never}\:{exceed}\:{the}\:{sphere}.\:\Rightarrow{OP}\leqslant{R}. \\ $$
Commented by mr W last updated on 13/Aug/21

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